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x [ n ] = eJnnn= cos R o n +j sin R o n in .NET framework
x [ n ] = eJnnn= cos R o n +j sin R o n Read QR Code 2d Barcode In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Quick Response Code Drawer In .NET Using Barcode generation for .NET Control to generate, create QR Code image in Visual Studio .NET applications. (1.53) Decode QR Code 2d Barcode In Visual Studio .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications. Barcode Generation In .NET Using Barcode encoder for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. Thus x [ n ] is a complex sequence whose real part is cos R o n and imaginary part is sin R o n .
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 1. CHAP. 11
SIGNALS AND SYSTEMS
Consider the complex exponential sequence with frequency (R, integer: ej(ilo+2rrk)n
+ 2 7 k ) , where k
is an
= e j R o nej2rrkn
 ei n o n  (1.56) since e j 2 " k n = 1. From Eq. (1.56) we see that the complex exponential sequence at frequency R, is the same as that at frequencies (R, f 27), (R, f 4.~1, and so on. Therefore, in dealing with discretetime exponentials, we need only consider an interval of length 2 7 in which to choose R,. Usually, we will use the interval 0 I R, < 2 7 or the interval  7 sr R, < 7 . General Complex Exponential Sequences: The most general complex exponential sequence is often defined as x [ n ] = Can
(1.57) where C and a are in general complex numbers. Note that Eq. (1.52) is the special case of Eq. (1.57) with C = 1 and a = eJRO. .. . Ylll
'111' = cos(rrn/6); Fig. 113 Sinusoidal sequences. (a) x[n] ( b )x[n] = cos(n/2). SIGNALS AND SYSTEMS
[CHAP. 1
Real Exponential Sequences: If C and a in Eq. (1.57) are both real, then x[n] is a real exponential sequence. Four distinct cases can be identified: a > 1 , 0 < a < 1,  1 < a < 0, and a <  1. These four real exponential sequences are shown in Fig. 112. Note that if a = 1, x[n] is a constant sequence, whereas if a =  1, x[n] alternates in value between + C and C. D. Sinusoidal Sequences: A sinusoidal sequence can be expressed as If n is dimensionless, then both R, and 0 have units of radians. Two examples of sinusoidal sequences are shown in Fig. 113. As before, the sinusoidal sequence in Eq. (1.58) can be expressed as As we observed in the case of the complex exponential sequence in Eq. (1.52), the same observations [Eqs. (1.54) and (1.5611 also hold for sinusoidal sequences. For instance, the sequence in Fig. 113(a) is periodic with fundamental period 12, but the sequence in Fig. l13( b ) is not periodic. SYSTEMS AND CLASSIFICATION OF SYSTEMS System Representation: A system is a mathematical model of a physical process that relates the input (or excitation) signal to the output (or response) signal. Let x and y be the input and output signals, respectively, of a system. Then the system is viewed as a transformation (or mapping) of x into y. This transformation is represented by the mathematical notation where T is the operator representing some welldefined rule by which x is transformed into y. Relationship (1.60) is depicted as shown in Fig. 114(a). Multiple input and/or output signals are possible as shown in Fig. 114(b). We will restrict our attention for the most part in this text to the singleinput, singleoutput case. System
Sy stem
Fig. 114 System with single or multiple input and output signals.
CHAP. 11
SIGNALS AND SYSTEMS
B. Continuous;Time and DiscreteTime Systems: If the input and output signals x and p are continuoustime signals, then the system is called a continuoustime system [Fig. I  15(a)].If the input and output signals are discretetime signals or sequences, then the system is called a discretetime s .stem [Fig. I  15(h)J.

